Magnetotactic bacteria in a droplet self-assemble into a rotary motor

Abstract

From intracellular protein trafficking to large-scale motion of animal groups, the physical concepts driving the self-organization of living systems are still largely unraveled. Self-organization of active entities, leading to novel phases and emergent macroscopic properties, recently shed new light on these complex dynamical processes. Here we show that under the application of a constant magnetic field, motile magnetotactic bacteria confined in water-in-oil droplets self-assemble into a rotary motor exerting a torque on the external oil phase. A collective motion in the form of a large-scale vortex, reversable by inverting the field direction, builds up in the droplet with a vorticity perpendicular to the magnetic field. We study this collective organization at different concentrations, magnetic fields and droplet radii and reveal the formation of two torque-generating areas close to the droplet interface. We characterize quantitatively the mechanical energy extractable from this new biological and self-assembled motor.

Fig. 1

Water-in-oil emulsion of magnetotactic bacteria. a A $\times 10$ phase-contrast image of an emulsion of magnetotactic bacteria (bacteria remain inside droplets) in hexadecane oil. A magnetic field of 4 mT is applied as indicated by the arrow (see the corresponding Supplementary Movie 1). A broad distribution of droplets radii is obtained, spanning typically from 20 to 120 µm. Zoom in: $\times 40$ phase-contrast image of two magnetotactic bacteria Magnetospirillum gryphiswaldense MSR-1 (darkest zones) swimming along the magnetic field direction. Zoom in: Sketch of a magnetotactic bacterium carrying magnetosomes (red squares) and two distal flagella. The magnetosomes are aligned along the body, generating a magnetic moment $m$. b Setup principle: a droplet, lying on the bottom plate of a pool of height $H=$ 270 µm and placed on the stage of an inverted microscope, is observed at its equatorial plane with a $\times 40$ objective. A uniform magnetic field is applied in the observation plane, parallel to the bottom and top plates of the pool. c Definitions of the north pole (NP), the south pole (SP) of the droplet and the spatial coordinates. $R$ is the droplet radius

Fig. 2

Influence of the cell density $n$ and the magnetic field $B$ on the emergence of collective vortical motion. a–c $\times 40$ phase-contrast images of droplets superimposed with time average PIV velocity fields (green arrows). We show the influence of the cell density $n$ on the phenomenology, the magnetic field is fixed $B=$ 4 mT. (a) $n~10^{14}$ bact m${}^{-3}$, $R=$ 43 µm: Bacteria accumulate at the poles of the droplet. (b) $n~10^{15}$ bact m${}^{-3}$, $R=$ 89 µm: unstable recirculation flows appear at the poles of the droplet. (c) $n~10^{17}$ bact m${}^{-3}$, $R=$ 55 µm: the bacteria self-organize to form a stable vortex flow at the center of the droplet. d–f Colored maps of the orthoradial projection of the instantaneous PIV velocity fields $V_{\theta }^{\mathrm{d}}$ (red-blue colormap, enhancing positive and negative values) superimposed with the instantaneous PIV velocity field (green arrows). The radius of the droplet is constant $R=$ 83 µm. We show the influence of the magnetic field magnitude on the phenomenology, the cell density is fixed $n=10^{17}$ bact m${}^{-3}$. d $B=$ 0.2 mT: no large scale collective motion is observed. e $B=$ 2 mT: vortex flow centered at the droplet center. f $B=$ 4 mT: the vortex flow is stronger than at $B=$ 2 mT. e, f Recirculation flows (negative values of $V_{\theta }^{\mathrm{d}}$) close to the poles are identified in blue

Fig. 3

Mechanical characterization of the rotary motor. a Mean orthoradial velocity profile $\overline{V_{\theta }^{\mathrm{d}}}\left(r\right)$ for one droplet of radius $R=$67 µm and for different magnetic field magnitudes $B$ (colors, from bottom to top $B=$ 1, 1.4, 2, 2.4, 3, 3.4, 4 mT). Close to the droplet core ($r=0$), the suspension rotates like a solid with a characteristic rotational velocity ${\mathrm{\Omega }}^{\mathrm{d}}$ which increases with $B$. The error bars are the standard errors. b Superposition of phase-contrast images (350 images corresponding to a 14 s movie) showing the circular rotation of the outer tracers for an inner rotational velocity ${\mathrm{\Omega }}^{\mathrm{d}}$= 0.13 rad s${}^{-1}$ measured at $B=$4 mT. c The torque $\tau$, acting on the oil and produced by the droplets, is extracted from the tracers orthoradial velocities (see (b)). We measure $\tau \left(B,R\right)$ for different droplets radii $R$ and magnetic field $B$ with respect to the core rotation velocity ${\mathrm{\Omega }}^{\mathrm{d}}\left(B,R\right)$ (average data for 10 droplets of similar radii, $\overline{R}$ is indicated by colors and is given with a $\pm 15$ µm standard deviation). The error bars are $\sigma ∕\sqrt{N}$ where $\sigma$ is the standard deviation and $N=10$ (number of droplets used for the average). d Torque by unit volume ${\tau }_v=\tau ∕\left(\frac{4}{3}\pi R^3\right)$ as a function of ${\mathrm{\Omega }}^{\mathrm{d}}$ for the same data set. The average data for different $\overline{R}$ collapse on the operating curve of the rotary motor. The velocity maps are the ones of the droplet displayed on Fig. 2d–f and placed at the corresponding operating points. The error bars are the standard errors

Fig. 4

Test of the scaling relation: $\tau =nmB\lambda \left(R,B\right)R^2$, where $\tau$ is the generated torque, $n~10^{17}$ bact m${}^{-3}$ is the bacteria density, $m~10^{-16}$ J T${}^{-1}$ is the magnetic moment of a single bacterium, $B$ is the magnetic field intensity and $\lambda \left(R,B\right)$ is a typical length inherent to the torque generation. a ${⟨\lambda ⟩}_B$ (resp. ${⟨\lambda ⟩}_R$) is the average value of $\lambda$ with respect to $B$ (resp. $R$). We only included the data corresponding to ${\tau }_v>0.1\mathrm{nN}\mathrm{\mu }{\mathrm{m}}^{-2}$, for which the torque is strong enough to be measured out from experimental noise. The error bars are the standard deviations of the average data. This graph shows that $\lambda =8\pm 2$ µm is an intrinsic length of the system which does not depend on $R$ nor $B$. The error bars are the standard errors. b Qualitative interpretation of the rotary motor self-organization. The volume of the recirculating bacteria contributing to the torque is dimensionally $\mathcal{V}~\lambda R^2$. The picture is similar close to the south pole of the droplet

Fig. 5

Vortex emergence and rotation reversal by magnetic field inversion. a Magnetic field amplitude $B$ as a function of time. b Response in the rotational velocity of the droplet core ${\mathrm{\Omega }}^{\mathrm{d}}$ under the applied magnetic field. The error bars are the standard errors. c, d, e The stages corresponding to the particular self-assemblies representations on the sketches on the right of the figure. The coral region corresponds to the central core rotation while the blue zones represent the recirculating bacteria. The bacterial suspension is dense ($n~10^{17}$ bact m${}^{-3}$). For each time step $t$, ${\mathrm{\Omega }}^{\mathrm{d}}\left(t\right)$ is computed from instantaneous PIV maps as in Fig. 3a. From the moment when the magnetic field is set on, ${\mathrm{\Omega }}^{\mathrm{d}}$ reaches a stationary value within a few seconds ($~$4 s). When reversing quickly the magnetic field direction while the suspension rotates CW, the suspension continues rotating CW at the short times after reversal before reversing completely its rotational direction to CCW. Then, ${\mathrm{\Omega }}^{\mathrm{d}}$ reaches a stable negative value $~$10 s after the magnetic field switch